This project is about bringing together ideas from three different areas of mathematics, namely algebra, mathematical physics, and number theory. It begins by considering some new objects in algebra, called cluster algebras, which have been around for just over a decade, and now constitute the most rapidly developing part of the subject. The project then looks within cluster algebras to find examples of discrete integrable systems, which are used by physicists to describe interactions between particles which evolve in discrete time steps. It then goes on to examine this evolution in discrete time from the viewpoint of number theory  dynamics over finite fields, to be precise  and ends up by showing how to generate number sequences with random behaviour from this discrete dynamics.
Some more details of the three different parts of the project are explained below.
I) Cluster algebras:
Most structures in algebra are defined by giving a set of objects, or generators, and then providing a set of relations or equations that they satisfy, which fix the rules for combining them. Cluster algebras are rather different, in that the generators are not given from the start, but are constructed in bunches (called clusters) from an initial set (called a seed) via a recursive process called mutation. The process of mutation is very complicated: at each step there are different paths to choose from, and generally the rules of mutation get changed along the way.
The inherent complexity of mutation is the main reason why the general structure of cluster algebras is not very well understood. However, in certain cases, the cluster algebra can end up looking the same after a finite number of mutation steps, up to symmetry. This periodic behaviour under sequences of mutations means that clusters can be produced by means of a single dynamical system, that is, a system which evolves in discrete time steps. The first main aim of the project is to classify the situations in which this periodicity arises.
II) Discrete integrable systems:
The laws of physics, such as Newton's laws of motion, are traditionally described mathematically in terms of differential equations, which describe how certain quantities vary smoothly over space and time. It was realized towards the end of the 19th century that, as soon as more than two particles interact, it is generally impossible to solve (or integrate) the laws of motion, and the behaviour of the system may even be chaotic. Nevertheless, exactly solvable (or integrable) systems helped to develop quantum theory, and enjoyed a renaissance in the second half of the 20th century when they were found in the description of stable solitary waves on shallow water, known as solitons.
The advent of soliton theory led to increased efforts to extend the properties of integrability to the discrete domain  both in space (evolution on a lattice), and in time (dynamics in discrete steps)  to produce discrete integrable systems. Very recently, I have found new examples of discrete integrable systems that appear inside cluster algebras with periodicity, and it is another main aim of my work to understand precisely when this happens.
III) Discrete dynamics over finite fields:
Imagine an unusual clock which is labelled with the numbers 1 up to p, where (unlike the number 12) p is a prime number, and in each hour the short hand moves from one number to the next. Starting from a given "time", the short hand will return to the same number every p hours. Thus one can add the numbers on the clock, forgetting any extra multiples of p. It turns out that one can also multiply, as well as subtract and divide (except by p), so that this "clock arithmetic" is a number system with p elements: the simplest example of a finite field.
I aim to show that, over finite fields, discrete integrable systems from cluster algebras give number sequences with exactly the properties required to make codes.
